The Continuum Limit and Positive Mass Gap in Four-Dimensional Yang Mills Theory.

This work is devoted to the mathematical study of four-dimensional quantum Yang–Mills theory and the problem of the existence of a mass gap (the Yang–Mills Mass Gap problem). The research examines the construction of gauge-invariant lattice measures, the transition from finite lattices to the continuum limit, the spectral properties of transfer operators, and the conditions required for the existence of a positive spectral gap. It also explores multiscale estimates, measure consistency, Poincaré inequalities, ground-state transformations, and the relationship between spectral gaps and exponential decay of correlation functions. The main objective is to bring these mathematical approaches together into a unified framework and investigate under what conditions the spectral properties of lattice theories can be preserved in the continuum limit. Artificial intelligence tools were used as auxiliary instruments during the preparation of this work, particularly for writing and formatting mathematical formulas, LaTeX editing, checking notation consistency, improving the presentation of mathematical arguments, and refining the text. AI tools did not replace the author's research efforts. The author takes responsibility for the mathematical content, statements, proofs, interpretations, and conclusions presented in this work.